Adaptive Estimation with Gaussian Radial Basis Mixtures

摘要

By considering a rich class of remodels with appropriately devised penalties, density estimators call be designed to naturally adapt to the complexity revealed by the data. This dissertation explores approximation, estimation, and computation properties of Gaussian mixtures to perform this type of adaptive estimation. For simplicity and clarity of exposition, we use equal weights and a fixed radial covariance, a model that we will call Gaussian radial basis mixtures (GRBMs). First, we generalize the MDL redundancy risk bound method of Barron and Cover (1991] to arbitrary penalties. Then we extend mixture redundancy bounds of Li [1999] to the case of unconstrained parameter spaces. These results together allow us to establish an exact risk bound for penalized likelihood GRBM estimation. Finally. simulations are performed to compare algorithms for optimizing the likelihood.